Question

Assume the​ zero-coupon yields on​ default-free securities are as summarized in the following​ table: Maturity 1 year 2 years 3 years 4 years 5 years ​Zero-Coupon Yields 3.30​% 3.80​% 4.00​% 4.40​% 4.50​% Consider a​ four-year, default-free security with annual coupon payments and a face value of $ 1 comma 000 that is issued at par. What is the coupon rate of this​ bond? The par coupon rate is nothing​%. ​(Round to two decimal​ places.)

Answer 1

Let the annual Coupon of the bond be $x

Tenure of the bond = 4 years.

Face Value of the bond = $1,000

Issue price of the bond =Present Value of the bond $1,000 (at par)

Let coupon amount of the bond be $x

.

Present Value of the Coupons at zero-coupon yields on​ default-free securities (computed in table below):

Years	Zero Coupon Yields	Discounting Factor at Yeild-Working	Discounting Factor at Yeild	Coupon Amount	Present Value at Yeildd
1	3.30%	1/1.033^1	0.968054211	x	0.968054211x
2	3.80%	1/1.038^2	0.928122482	x	0.928122482x
3	4%	1/1.04^3	0.888996359	x	0.888996359x
4	4.40%	1/1.044^4	0.841778841	x	0.841778841x
			3.626951894		3.626951894x

.

Present Value of the bond = Present Value of the Coupons at zero-coupon yields on​ default-free securities + Present Value of the Redemption Value at zero-coupon yields on​ default-free securities

1,000 = 3.626951894x + 1,000* 0.841778841

1,000 = 3.626951894x + 841.7788415

3.626951894x = 158.22

x = 158.22/3.626951894

x = 43.62

.

Therefore, annual coupon amount of the bond = $43.62

.

Therefore, Coupon Rate of the Bond = Annual coupon amount of the bond/Face Value of the Bond * 100

= 43.62/1,000 * 100

= 4.63%